Eter gaussian and delta correlated in space s 0 s 2 δ

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eter, Gaussian and delta-correlated in space:[s] = 0,[s2] = Δσ2(27)The above equation must be solved with a (conventional) uniform initial conditiong(t)1/taat small times. The simplified SBR correlator is obtained as the averagesolution of the stochastic equation over the random fluctuations:7
July 24, 2015Philosophical Magazineglass-FAIgsSBR(σ, t)[g(t)](28)The simplified SBR equation is nothing but a version of the full SBR equationspresented earlier with no space dependence. The model in itself possess a numberof interesting properties discussed in [2] and more exstensively in [3]. With abovethe definition we can state the following [2]For all i the dynamical coefficientsg±i(t)are the same of the asymptotic ex-pansion ofgsSBR(t):gscalτ, t)gsSBRτ, t)(29)Note that while for the full SBR equations we cannot exhibit an explicit solutionand we must resort to numerics, for the simplified model we can write down theexplicit solution in terms of the functionsg±0(t):GsSBR(σ, t) =integraldisplay+−∞ds2πΔσe(s-σ)2σ2|s|1/2gsign[s]0(t|s|1/2a),(30)and in order to have an idea of the complexity of the coefficientsg±i(t) of thefinite-volume expansion the reader can reverse-engineer them by computing theasymptotic expansion for large absolute values ofσ(or more precisely for largeabsolute values ofσ/Δσ) by using the representation12πΔσe(s-σ)2σ2=δ(sσ) +Δσ22δ′′(sσ) +. . .(31)Note that the expansion for large positive or negative values ofσwould be an ex-pansion aroundg+0(t) org0(t) respectively. At this stage it should be clear that SBRis not just a nice phenomenological model. The problem of finite-size corrections inSG is an example where one can derive it from first principles and verify its validityprovided the scaling quantity2is small enough. Later we will discuss that thisderivation actually depends on the fact that the finite-size corrections depends onan effective Landau theory (the GCT mentioned in the introduction) and it is thistheory that it is actually equivalent perturbatively to SBR. On the other hand therelevance of GCT (and thus the mapping to SBR) goes well beyond the problemof finite-size corrections in mean-field models and allows to discuss SG in physicaldimension and supercooled liquids.We also note once again that while the random fluctuations of the temperaturein thestatictreatment leads to the ill-defined expression [˜τ+h] the random fluc-tuations in the dynamic treatment leads to expression (30) which is well defined asa function of time andσ.2.2.Finite-Dimensional Spin-Glass Beyond Mean-FieldIn the previous subsection we have explained how some physical quantities (theexpansion coefficientsg±i(t)) that we can compute explicitly in some Spin-Glassmodels can be also computed from the asymptotic expansion of SBR. Furthermorewe are not really interested in the expansion coefficients themselves but rather in8
July 24, 2015Philosophical Magazineglass-FAItheir resummation and therefore we can forget completely the expansion and takeSBR as the scaling theory valid near the critical temperature.

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Term
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Critical Point, Critical phenomena, Renormalization group, Philosophical Magazine, SBR

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